There is no formula/equation to convert from the PLSS to lat/lon. The PLSS is a reference grid, not a coordinate system (and note that it doesn't cover the entire US). To automate this, you would need to generate (or find - I don't know if one is out there) a coordinate lookup table for a script to reference or to use in a join.
Your lookup table would need to have a common field with your data (which may require modifying said data). At a minimum this would be the township-range-section-qq, but there's an added complication. Township and range are specified from reference baselines and meridians. In other words, there is more than one Township 1 North, Range 1 West. The example you posted doesn't give any method of identifying which baseline/meridian to use, and if you're covering such a large area you will most certainly run into more than one of them. One you had this table with every qq's centroid by name/reference in one column and the lat/lon for it in another (or others), you can join or add those coordinates to your data and then plot them.
As you've seen, it is possible to find the lat/lon of the corners/centroid for a given grid reference. From looking at your data, you would need to do this at the QQ (quarter-quarter) level. For many, but not all, areas covered by the PLSS, you can download a grid dataset to generate centroids and then coordinates from. But there's no single source for the entire system, and not all sources go past the section level. You'll also end up with multiple points in the same location, since anything in the same qq will have the same coordinate.
Alternatively you would have to find a service out there that can do this lookup for you. You've already found some tools that will do it one point at a time, but I believe some of them are also services (such as the BLM link in my comment on Dana's post). You have to learn the API and how to format your data for input, but it can be done in batch and not one at a time. Be aware most services like this have quantity limits that are either time or cost based.